How do you stage an exhibition of a painter who, within a few years of launching his career, announced that “painting is long obsolete and the painter himself a prejudice of the past”? Since it opened, Tate Modern has sought to mount a Kazimir Malevich retrospective. No painter is a more significant influence on the 20th-century trajectory – abstraction, minimalism, conceptualism – that the gallery strives to define. At the same time, none is more elusive, mysterious, unpredict­able or contradictory.

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Dispensing with imagery, colour or composition, “Black Square”, now 99 years old, dismantled painting’s historic parameters. Malevich intended it as an avant-garde icon, superseding Russian Orthodox icons, and so it became. But this masterpiece is too fragile to leave Moscow, and many of the suprematist canvases launched with it – fiercely purist designs in black, white and unmixed primary colours that declared the end of pictorial representation – are little known in the west. In Russia, Malevich was not exhibited between his death in 1935 and 1988; only in the last two decades have his experiments and elaboration of what he called “the zero of form” been fully explored.

Tate’s marvellously orchestrated, deeply revealing show builds on results of that research and benefits from stellar worldwide loans. Exuberantly varied, it places the suprematist adventure in dazzlingly unexpected contexts. It opens with Malevich’s impressionist pieces dissolving light and colour, as in the ochre dome glittering behind snow-laden trees in “Church” (1905), fresh as a winter walk in the woods, and symbolist works such as a satanic “Self-portrait” (1909), whose frontality, brilliant crimson, curvilinear stylisation and insistence on artist as seer all owe debts to Van Gogh. It ends with the poignant, still controversial “Second Peasant Cycle”, never before displayed in the UK.

One gallery is devoted to works on paper from the outstanding Nicholas Khardiev collection, which demonstrate how meticulously Malevich forged a language to respond to radical political impulse. Another recreates the UNOVIS workshop in Vitebsk where Malevich was so mesmerising a teacher-theorist that his designs – green circles, orange squares, blue triangles – were pasted across town, one commentator remembered, like “suprematist confetti”. A third gallery displays blocky stage and costume designs, and a film recreation of Victory over the Sun, Malevich’s and futurist poet Velimir Khlebnikov’s 1913 opera. Its jangling discordant strains ring out across the show, relating how “future men” tame the sun, pull it down from heaven, and plunge the world into darkness to prepare for the advent of new light.

That cosmic drama underpins suprematism, whose sweeping rise as the house style of the Bolsheviks reflected longing for a new world. Unlike his contemporaries – Kandinsky, Chagall – Malevich by 1915 had never left Russia (he travelled abroad only once, in 1927), and although he knew impressionism and cubism from Ivan Morozov’s and Sergei Shchukin’s Moscow collections, his Russian identity was untamed by western assimilation. “Peasant Woman with Buckets and Child” (1912) and “The Scyther” (1912), his initial responses to cubism, are rooted in images of prehistoric Russia, the massive bodies and heavy features recalling primitive Slavic stone idols.

The mower’s tubular form and metallic sheen suggest mechanical and revolutionary power; he stands symbolically against red mounds of earth. His successor, “The Woodcutter” (1912), is integrated into a ground of logs simplified into abstract cylinders, which the woodcutter chops up. Malevich, reared in rural Ukraine, identified with the peasant: his woodcutter is a visionary poet-painter, breaking up object and word to reveal truth beyond appearance. “We have cut the object! We have begun to see through the world!” boasted Malevich’s poet-friend Aleksei Kruchenykh.

From 1913 to 1914, Malevich played cat and mouse with concealment and disclosure in variations of analytic cubism. “Samovar” emerges in a series of silver planes, dark curves referencing wooden knobs and handles. The heroine of “Woman at a Tram Stop” is invisible in the overlapping masses abstracting tram lines, wheels and timetable.

It is a step now to “Suprematist Composition: Airplane Flying” (1915): black rectangles increasing in size from right to left, followed by yellow rectangles moving left to right, allude to a climbing plane, seen from the air as it crosses a tipped red horizon line. There is no mimetic illustration, but we are caught in sensations of lift and flight here.

And then, through civil war and repression, he fell silent as a painter for a decade. He returned, in 1928, with monumental depictions of hieratic, faceless figures with cross-like heads and bodies divided into strict vertical colour areas: “Head of a Peasant”, “Harvesting”, “Woman with a Rake”, “Sportsmen”. Motionless as saints in Russian icons, these emblems of suffering and redemption were painted during the horrors of famine and farm collectivisation. By then Malevich was out of favour, but this return to the human form was far from a compromise with socialist realism; rather, Malevich integrates suprematism into an urgent hybrid language crossing abstraction and figuration.

Self-depictions abound here: symbolic, as in “Female Torso” (1933), where the powerless Malevich identifies with an armless peasant; or playing naturalism against artifice in Moscow’s 1933 “Self-portrait”, a conscious echo of the assertive Renaissance pose in Dürer’s 1500 self-portrait but with a collar of red-and-white triangles a final suprematist allusion.

This shows an artist at once harsh and tender, radical and in love with art history, denying individual expression yet poignantly autobiographical and, as he confronts us with a defiant, warm stare, surely prescient of the complex journey that his abstraction would eventually make out into the world.